Theorem suplocexprlemlub 7850

Description: Lemma for suplocexpr 7851. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemlub	⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑥,𝐴,𝑦   𝑧,𝐵   𝜑,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑤,𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦<P 𝐵)
2		ltrelpr 7631	. . . . . . . 8 ⊢ <P ⊆ (P × P)
3	2	brel 4732	. . . . . . 7 ⊢ (𝑦<P 𝐵 → (𝑦 ∈ P ∧ 𝐵 ∈ P))
4	3	simpld 112	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝑦 ∈ P)
5	4	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦 ∈ P)
6	3	simprd 114	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝐵 ∈ P)
7	6	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝐵 ∈ P)
8		ltdfpr 7632	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝐵 ∈ P) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
9	5, 7, 8	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
10	1, 9	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))
11		simprrr 540	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (1st ‘𝐵))
12		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
13	12	fveq2i 5589	. . . . . . . . 9 ⊢ (1st ‘𝐵) = (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
14		npex 7599	. . . . . . . . . . . . 13 ⊢ P ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → P ∈ V)
16		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
17		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
18		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
19	16, 17, 18	suplocexprlemss 7841	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
20	15, 19	ssexd 4189	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
21		fo1st 6253	. . . . . . . . . . . . 13 ⊢ 1st :V–onto→V
22		fofun 5508	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
24		funimaexg 5364	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V)
25	23, 24	mpan 424	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (1st “ 𝐴) ∈ V)
26		uniexg 4491	. . . . . . . . . . 11 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V)
27	20, 25, 26	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (1st “ 𝐴) ∈ V)
28		nqex 7489	. . . . . . . . . . 11 ⊢ Q ∈ V
29	28	rabex 4193	. . . . . . . . . 10 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V
30		op1stg 6246	. . . . . . . . . 10 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
31	27, 29, 30	sylancl 413	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
32	13, 31	eqtrid 2251	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐵) = ∪ (1st “ 𝐴))
33	32	eleq2d 2276	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
34	33	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
35	11, 34	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ ∪ (1st “ 𝐴))
36		suplocexprlemell 7839	. . . . 5 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
37	35, 36	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
38		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ Q)
39	38	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ Q)
40		simprrl 539	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (2nd ‘𝑦))
41	40	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (2nd ‘𝑦))
42		simpr 110	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (1st ‘𝑧))
43		rspe 2556	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
44	39, 41, 42, 43	syl12anc 1248	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
45	4	ad4antlr 495	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦 ∈ P)
46	19	ad4antr 494	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝐴 ⊆ P)
47		simplr 528	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ 𝐴)
48	46, 47	sseldd 3196	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ P)
49		ltdfpr 7632	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
50	45, 48, 49	syl2anc 411	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
51	44, 50	mpbird 167	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦<P 𝑧)
52	51	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) → (𝑠 ∈ (1st ‘𝑧) → 𝑦<P 𝑧))
53	52	reximdva 2609	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
54	37, 53	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
55	10, 54	rexlimddv 2629	. 2 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
56	55	ex 115	1 ⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489  Vcvv 2773   ⊆ wss 3168  ⟨cop 3638  ∪ cuni 3853  ∩ cint 3888   class class class wbr 4048   “ cima 4683  Fun wfun 5271  –onto→wfo 5275  ‘cfv 5277  1^st c1st 6234  2^nd c2nd 6235  Qcnq 7406   <_Q cltq 7411  Pcnp 7417  <_P cltp 7421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-1st 6236  df-qs 6636  df-ni 7430  df-nqqs 7474  df-inp 7592  df-iltp 7596

This theorem is referenced by:  suplocexpr  7851